Question

if cos ^-1p+cos^-1q+cos^-1r=π, then find value of p^2+q^2+r^2+2pqr.​

Answer 1

Step-by-step explanation:

Given, cos

−1

p+cos

−1

q+cos

−1

r=π

⇒cos

−1

(pq−

(1−p

2

)(1−q

2

)

)=π−cos

−1

r

⇒cos

−1

(pq−

(1−p

2

)(1−q

2

)

)=

2

π

+sin

−1

r

⇒sin

−1

(pq−

(1−p

2

)(1−q

2

)

)=sin

−1

(−r)

⇒pq−

(1−p

2

)(1−q

2

)

=−r

⇒pq+r=

(1−p

2

)(1−q

2

)

⇒p

2

q

2

+2pqr+r

2

=1−p

2

−q

2

+p

2

q

2

⇒p

2

+q

2

+r

2

+2pqr=1